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+ Prob CFt=1 will be case 2 В· CFt=1 cash п¬‚ow in case 2

+ Prob CFt=1 will be case 3 В· CFt=1 cash п¬‚ow in case 3 .

This expected return of \$206.80 is less than the \$210 that the government promises. Put
diп¬Ђerently, if I promise you a rate of return of 5%,
\$210 в€’ \$200
Promised(Лњt=0,1 ) = = 5.00%
r
\$200
(5.11)
Лњ
Promised(Ct=1 ) в€’ CFt=0
Promised(Лњt=0,1 ) =
r ,
CFt=0

your expected rate of return is only
\$206.80 в€’ \$200
E (Лњt=0,t=1 ) = = 3.40%
r
\$200
(5.12)
Лњ
E (Ct=1 ) в€’ CFt=0
E (Лњt=0,t=1 ) =
r ,
CFt=0

which is less than the 5% interest rate that Uncle Sam promisesвЂ”and surely delivers.
You need to determine how much I have to promise you to вЂњbreak even,вЂќ so that you expect to Determine how much
more interest promise
end up with the same \$210 that you could receive from Uncle Sam. In Table 5.1, we computed
you need to break even.
the expected payoп¬Ђ as the probability-weighted average payoп¬Ђ. You want this payoп¬Ђ to be not
п¬Ѓle=uncertainty.tex: LP
90 Chapter 5. Uncertainty, Default, and Risk.

\$206.80, but the \$210 that you can get if you put your money in government bonds. So, you
now solve for an amount x that you want to receive if I have money,

Лњ
E (Ct=1 ) = В· x
98%

+ В·
1% \$100

+ В· = \$210.00
1% \$0
(5.13)
Лњ
E (Ct=1 ) = Prob CFt=1 will be case 1 В· CFt=1 cash п¬‚ow in case 1

+ Prob CFt=1 will be case 2 В· CFt=1 cash п¬‚ow in case 2

+ Prob CFt=1 will be case 3 В· CFt=1 cash п¬‚ow in case 3 .

The solution is that if I promise you x = \$213.27, you will expect to earn the same 5% interest
rate that you can earn in Treasury bonds. Table 5.2 conп¬Ѓrms that a promise of \$213.27 for a
cash investment of \$200, which is a promised interest rate of
\$213.27 в€’ \$200
Promised(Лњt=0,1 ) = = 6.63%
r
\$200
(5.14)
Лњ
Promised(Ct=1 ) в€’ CFt=0
Promised(Лњt=0,1 ) =
r ,
CFt=0

and provides an expected interest rate of

(5.15)
E (Лњt=0,1 ) = 98% В· (+6.63%) + 1% В· (в€’50%) + 1% В· (в€’100%) = 5% .
r

Table 5.2. Risky Payoп¬Ђ Table: 6.63% Promised Interest Rate

How Likely
Flow CFt=1 return of (Probability)
\$213.27 +6.63% 98% of the time
\$100.00 вЂ“50.00% 1% of the time
\$0.00 вЂ“100.00% 1% of the time
\$210.00 +5.00% in expectation

The diп¬Ђerence of 1.63% between the promised (or quoted) interest rate of 6.63% and the ex-
The difference between
the promised and pected interest rate of 5% is the default premiumвЂ”it is the extra interest rate that is caused by
expected interest rate is
the default risk. Of course, you only receive this 6.63% if everything goes perfectly. In our
perfect world with risk-neutral investors,

= +
6.63% 5% 1.63%
(5.16)

Important: Except for 100% safe bonds (Treasuries), the promised (or quoted)
rate of return is higher than the expected rate of return. Never confuse the higher
promised rate for the lower expected rate.
Financial securities and information providers rarely, if ever, provide expected
rates of return.
п¬Ѓle=uncertainty.tex: RP
91
Section 5В·2. Interest Rates and Credit Risk (Default Risk).

On average, the expected rate of return is the expected time premium plus the expected default In a risk-neutral world,
all securities have the
same exp. rate of return.

E Rate of Return = E Time Premium + E Realized Default Premium
(5.17)
= E Time Premium + .
0

If you want to work this out, you can compute the expected realized default premium as follows:
you will receive (6.63% в€’ 5% = 1.63%) in 98% of all cases; в€’50% в€’ 5% = в€’55% in 1% of all cases
(note that you lose the time-premium); and в€’100% в€’ 5% = в€’105% in the remaining 1% of all
cases (i.e., you lose not only all your money, but also the time-premium). Therefore,

(5.18)
E Realized Default Premium = 98% В· (+1.63%) + 1% В· (в€’55%) + 1% В· (в€’105%) = 0% .

Solve Now!
Q 5.6 Recompute the example in Table 5.2 assuming that the probability of receiving full pay-
ment of \$210 is only 95%, the probability of receiving \$100 is 1%, and the probability of receiving
absolutely no payment is 4%.

(a) At the promised interest rate of 5%, what is the expected interest rate?

(b) What interest rate is required as a promise to ensure an expected interest rate of 5%?

5В·2.C. Preview: Risk-Averse Investors Have Demanded Higher Expected Rates

We have assumed that investors are risk-neutralвЂ”indiп¬Ђerent between two loans that have the In addition to the
same expected rate of return. As we have already mentioned, in the real world, risk-averse
life, investors also
investors would demand and expect to receive a little bit more for the risky loan. Would you demand a risk premium.
rather invest into a bond that is known to pay oп¬Ђ 5% (for example, a U.S. government bond), or
would you rather invest in a bond that is вЂњmerelyвЂќ expected to pay oп¬Ђ 5% (such as my 6.63%
bond)? Like most lenders, you are likely to be better oп¬Ђ if you know exactly how much you will
receive, rather than live with the uncertainty of my situation. Thus, as a risk-averse investor,
you would probably ask me not only for the higher promised interest rate of 6.63%, which only
gets you to an expected interest rate of 5%, but an even higher promise in order to get you more
than 6.63%. For example, you might demand 6.75%, in which case you would expect to earn
not just 5%, but a little more. The extra 12 basis points is called a risk premium, and it is an
interest component required above and beyond the time premium (i.e., what the U.S. Treasury
Department pays for use of money over time) and above and beyond the default premium (i.e.,
what the promised interest has to be for you to just expect to receive the same rate of return
as what the government oп¬Ђers).
Recapping, we know that 5% is the time-value of money that you can earn in interest from the A more general
decomposition of rates
Treasury. You also know that 1.63% is the extra default premium that I must promise you, a
of return.
risk-neutral lender, to allow you to expect to earn 5%, given that repayment is not guaranteed.
Finally, if you are not risk-neutral but risk-averse, I may have to pay even more than 6.63%,
although we do not know exactly how much.
If you want, you could think of further interest decompositions. It could even be that the time- More intellectually
interesting, but
premium is itself determined by other factors (such as your preference between consuming
otherwise not too useful
today and consuming next year, the inп¬‚ation rate, taxes, or other issues, that we are brushing decompositions.
over). Then there would be a liquidity premium, an extra interest rate that a lender would
demand if the bond could not easily be soldвЂ”resale is much easier with Treasury bonds.
п¬Ѓle=uncertainty.tex: LP
92 Chapter 5. Uncertainty, Default, and Risk.

Important: When repayment is not certain, lenders demand a promised interest
rate that is higher than the expected interest rate by the default premium.

Promised Interest Rate
(5.19)

The promised default premium is positive, but it is only paid when everything goes
well. The actually earned interest rate consists of the time premium, the realized
risk premium, and a (positive or negative) default realization.

Actual Interest Rate Earned
(5.20)

The default realization could be more than negative enough to wipe out both the
time premium and the risk premium. But it is zero on average. Therefore,

Expected Interest Rate
(5.21)

The risk premium itself depends on such strange concepts as the correlation of loan default
Some real world
evidence. with the general economy and will be the subject of Part III of the book. However, we can preview
the relative importance of these components for you in the context of corporate bonds. (We
will look at risk categories of corporate bonds in more detail in the next chapter.) The highest-
quality bonds are called investment-grade. A typical such bond may promise about 6% per
annum, 150 to 200 basis points above the equivalent Treasury. The probability of default
would be smallвЂ”less than 3% in total over a ten-year horizon (0.3% per annum). When an
investment-grade bond does default, it still returns about 75% of what it promised. For such
bonds, the risk premium would be smallвЂ”a reasonable estimate would be that only about 10
to 20 basis points of the 200 basis point spread is the risk premium. The quoted interest rate
of 6% per annum therefore would reп¬‚ect п¬Ѓrst the time premium, then the default premium,
and only then a small risk premium. (In fact, the liquidity premium would probably be more
important than the risk premium.) For low-quality corporate bonds, however, the risk premium
can be important. Ed Altman has been collecting corporate bond statistics since the 1970s. In
an average year, about 3.5% to 5.5% of low-grade corporate bonds defaulted. But in recessions,
the default rate shot up to 10% per year, and in booms it dropped to 1.5% per year. The average
value of a bond after default was only about 40 cents on the dollar, though it was as low 25
cents in recessions and as high as 50 cents in booms. Altman then computes that the most
risky corporate bonds promised a spread of about 5%/year above the 10-Year Treasury bond,
but ultimately delivered a spread of only about 2.2%/year. 280 points are therefore the default
premium. The remaining 220 basis points contain both the liquidity premium and the risk
Solve Now!
Q 5.7 Return to the example in Table 5.2. Assume that the probability of receiving full payment
of \$210 is only 95%, the probability of receiving \$100 is 4%, and the probability of receiving
absolutely no payment is 1%. If the bond quotes a rate of return of 12%, what is the time premium,
п¬Ѓle=uncertainty.tex: RP
93
Section 5В·3. Uncertainty in Capital Budgeting, Debt, and Equity.

5В·3. Uncertainty in Capital Budgeting, Debt, and Equity

We now turn to the problem of selecting projects under uncertainty. Your task is to compute
present values with imperfect knowledge about future outcomes. Your principal tool in this
task will be the payoп¬Ђ table (or state table), which assigns probabilities to the project value in
each possible future value-relevant scenario. For example, a п¬‚oppy disk factory may depend
on computer sales (say, low, medium, or high), whether п¬‚oppy disks have become obsolete
(yes or no), whether the economy is in a recession or expansion, and how much the oil price
(the major cost factor) will be. Creating the appropriate state table is the managerвЂ™s taskвЂ”
judging how the business will perform depending on the state of these most relevant variables.
Clearly, it is not an easy task even to think of what the key variables are, to determine the
probabilities under which these variables will take on one or another value. Assessing how
your own project will respond to them is an even harder taskвЂ”but it is an inevitable one. If
you want to understand the value of your project, you must understand what the projectвЂ™s key
value drivers are and how the project will respond to these value drivers. Fortunately, for many
projects, it is usually not necessary to describe possible outcomes in the most minute detailвЂ”
just a dozen or so scenarios may be able to cover the most important information. Moreover,
these state tables will also allow you to explain what a loan (also called debt or leverage) and
levered ownership (also called levered equity) are, and how they diп¬Ђer.

5В·3.A. Present Value With State-Contingent Payoп¬Ђ Tables

Almost all companies and projects are п¬Ѓnanced with both debt and levered equity. We already Most projects are
п¬Ѓnanced with a mix of
know what debt is. Levered equity is simply what accrues to the business owner after the debt
debt and equity.
is paid oп¬Ђ. (In this chapter, we shall not make a distinction between п¬Ѓnancial debt and other
house with a mortgage, you really own the house only after you have made all debt payments.
If you have student loans, you yourself are the levered owner of your future income stream.
That is, you get to consume вЂњyourвЂќ residual income only after your liabilities (including your
non-п¬Ѓnancial debt) are paid back. But what will the levered owner and the lender get if the
companyвЂ™s projects fail, if the house collapses, or if your career takes a turn towards Rikers
Island? What is the appropriate compensation for the lender and the levered owner? The split
of net present value streams into loans (debt) and levered equity lies at the heart of п¬Ѓnance.
We will illustrate this split through the hypothetical purchase of a building for which the fu- The example of this
section: A building in
ture value is uncertain. This building is peculiar, though: it has a 20% chance that it will be
destroyed, say by a tornado, by next year. In this case, its value will only be the landвЂ”say, one of two possible
\$20,000. Otherwise, with 80% probability, the building will be worth \$100,000. Naturally, the future values.
\$100,000 market value next year would itself be the result of many factorsвЂ”it could include
any products that have been produced inside the building, real-estate value appreciation, as
well as a capitalized value that takes into account that a tornado might strike in subsequent
years.

Table 5.3. Building Payoп¬Ђ Table

Event Probability Value
Sunshine 80% \$100,000
Expected Future Value \$84,000
п¬Ѓle=uncertainty.tex: LP
94 Chapter 5. Uncertainty, Default, and Risk.

The Expected Building Value
Table 5.3 shows the payoп¬Ђ table for full building ownership. The expected future building
To obtain the expected
future cash value of the value of \$84,000 was computed as
building, multiply each
(possible) outcome by its
probability. E (Valuet=1 ) = В·
20% \$20, 000

+ В· = \$84, 000
80% \$100, 000
(5.22)

+ Prob( Sunshine ) В· (Value if Sunshinet=1 ) .

Now, assume that the appropriate expected rate of return for a project of type вЂњbuildingвЂќ with
Then discount back the
expected cash value this type of riskiness and with one-year maturity is 10%. (This 10% discount rate is provided by
using the appropriate
demand and supply in the п¬Ѓnancial markets and known.) Your goal is to determine the present
cost of capital.
valueвЂ”the appropriate correct priceвЂ”for the building today.
There are two methods to arrive at the present value of the buildingвЂ”and they are almost
Under uncertainty, use
NPV on expected (rather identical to what we have done earlier. We only need to replace the known value with the
than actual, known) cash
expected value, and the known future rate of return with an expected rate of return. Now, the
п¬‚ows, and use the
п¬Ѓrst PV method is to compute the expected value of the building next period and to discount
appropriate expected
(rather than actual,
it at the cost of capital, here 10 percent,
known) rates of return.
The NPV principles
\$84, 000
remain untouched. PVt=0 = в‰€ \$76, 363.64
1 + 10%
(5.23)
E (Valuet=1 )
= .
1 + E (rt=0,1 )

Table 5.4. Building Payoп¬Ђ Table, Augmented

в‡’
Event Probability Value Discount Factor PV
в‡’
1/(1+10%)
в‡’
1/(1+10%)
Sunshine 80% \$100,000 \$90,909.09

The second method is to compute the discounted state-contingent value of the building, and
Taking expectations and
discounting can be done then take expected values. To do this, augment Table 5.3. Table 5.4 shows that if the tornado
in any order.
strikes, the present value is \$18,181.82. If the sun shines, the present value is \$90,909.10.
Thus, the expected value of the building can also be computed

PVt=0 = В·
20% \$18, 181.82

+ В· в‰€ \$76, 363.64
80% \$90, 909.09
(5.24)

+ Prob( Sunshine ) В· (PV of Building if Sunshine) .

Both methods lead to the same result: you can either п¬Ѓrst compute the expected value next year
(20% В· \$20, 000 + 80% В· \$100, 000 = \$84, 000), and then discount this expected value of \$84,000
to \$76,363.34; or you can п¬Ѓrst discount all possible future outcomes (\$20,000 to \$18,181.82;
and \$100,000 to \$90,909.09), and then compute the expected value of the discounted values
(20% В· \$18, 181.82 + 80% В· \$90, 909.09 = \$76, 363.34.)
п¬Ѓle=uncertainty.tex: RP
95
Section 5В·3. Uncertainty in Capital Budgeting, Debt, and Equity.

Important: Under uncertainty, in the NPV formula,

вЂў known future cash п¬‚ows are replaced by expected discounted cash п¬‚ows, and
вЂў known appropriate rates of return are replaced by appropriate expected
rates of return.

You can п¬Ѓrst do the discounting and then take expectations, or vice-versa.

The State-Dependent Rates of Return
What would the rates of return be in both states, and what would the overall expected rate of The state-contingent
rates of return can also
return be? If you have bought the building for \$76,363.64, and no tornado strikes, your actual
be probability-weighted
rate of return (abbreviated rt=0,1 ) will be to arrive at the average
(expected) rate of
\$100, 000 в€’ \$76, 363.64 return.
rt=0,1 = в‰€ +30.95% . (5.25)
if Sunshine:
\$76, 363.64

\$20, 000 в€’ \$76, 363.64
rt=0,1 = в‰€ в€’73.81% . (5.26)
\$76, 363.64

Therefore, your expected rate of return is

E (Лњt=0,1 ) = В·
r (в€’73.81%)
20%

+ В· = 10.00%
(+30.95%)
80%
(5.27)
r

+ Prob( Sunshine ) В· (rt=0,1 if Sunshine) .

The probability state-weighted rates of return add up to the expected overall rate of return.
This is as it should be: after all, we derived the proper price of the building today using a 10%
expected rate of return.
Solve Now!
Q 5.8 What changes have to be made to the NPV formula to handle an uncertain future?

Q 5.9 Under risk-neutrality, a factory can be worth \$500,000 or \$1,000,000 in two years, de-
pending on product demand, each with equal probability. The appropriate cost of capital is 6%
per year. What is the present value of the factory?

Q 5.10 A new product may be a dud (20% probability), an average seller (70% probability) or
dynamite (10% probability). If it is a dud, the payoп¬Ђ will be \$20,000; if it is an average seller, the
payoп¬Ђ will be \$40,000; if it is dynamite, the payoп¬Ђ will be \$80,000. What is the expected payoп¬Ђ
of the project?

Q 5.11 (Continued.) The appropriate expected rate of return for such payoп¬Ђs is 8%. What is the
PV of the payoп¬Ђ?

Q 5.12 (Continued.) If the project is purchased for the appropriate present value, what will be
the rates of return in each of the three outcomes?

Q 5.13 (Continued.) Conп¬Ѓrm the expected rate of return when computed from the individual
outcome speciп¬Ѓc rates of return.
п¬Ѓle=uncertainty.tex: LP
96 Chapter 5. Uncertainty, Default, and Risk.

5В·3.B. Splitting Project Payoп¬Ђs into Debt and Equity

We now know how to compute the NPV of state-contingent payoп¬ЂsвЂ”our building paid oп¬Ђ dif-
State-contingent claims
have payoffs that ferently in the two states of nature. Thus, our building was a state-contingent claimвЂ”its payoп¬Ђ
depend on future states
depended on the outcome. But it is just one of many. Another state-contingent claim might
of nature.
promise to pay \$1 if the sun shines and \$25 if a tornado strikes. Using payoп¬Ђ tables, we can
work out the value of any state-contingent claimsвЂ”and in particular the value of the two most
important state-contingent claims, debt and equity.

The Loan
We now assume you want to п¬Ѓnance the building purchase of \$76,363.64 with a mortgage of
Assume the building is
funded by a mortgagor \$25,000. In eп¬Ђect, the single project вЂњbuildingвЂќ is being turned into two diп¬Ђerent projects,
and a residual, levered
each of which can be owned by a diп¬Ђerent party. The п¬Ѓrst project is the project вЂњMortgage
building owner.
Lending.вЂќ The second project is the project вЂњResidual Building Ownership,вЂќ i.e., ownership of
the building but bundled with the obligation to repay the mortgage. This вЂњResidual Building
OwnershipвЂќ investor will not receive a dime until after the debt has been satisп¬Ѓed. Such residual
ownership is called the levered equity, or just the equity, or even the stock in the building, in
order to avoid calling it вЂњwhatвЂ™s-left-over-after-the-loans-have-been-paid-oп¬Ђ.вЂќ
What sort of interest rate would the creditor demand? To answer this question, we need to
The п¬Ѓrst goal is to
determine the know what will happen if the building were to be condemned, because the mortgage value
appropriate promised
(\$25,000 today) will be larger than the value of the building if the tornado strikes (\$20,000
interest rate on a
next year). We are assuming that the owner could walk away from it and the creditor could
вЂњ\$25,000 value todayвЂќ
mortgage loan on the
repossess the building, but not any of the borrowerвЂ™s other assets. Such a mortgage loan is
building.
called a no-recourse loan. There is no recourse other than taking possession of the asset itself.
This arrangement is called limited liability. The building owner cannot lose more than the
money that he originally puts in. Limited liability is a mainstay of many п¬Ѓnancial securities: for
example, if you purchase stock in a company in the stock market, you cannot be held liable for

Table 5.5. Payoп¬Ђ to Mortgage Creditor, Providing \$25,000 Today

Prob
Event Value Discount Factor
1/(1+10%)
1/(1+10%)
Sunshine 80% Promised

Anecdote:
The framers of the United States Constitution had the English bankruptcy system in mind when they included
the power to enact вЂњuniform laws on the subject of bankruptciesвЂќ in the Article I Powers of the legislative branch.
The п¬Ѓrst United States bankruptcy law, passed in 1800, virtually copied the existing English law. United States
bankruptcy laws thus have their conceptual origins in English bankruptcy law prior to 1800. On both sides of
the Atlantic, however, much has changed since then.
Early English law had a distinctly pro-creditor orientation, and was noteworthy for its harsh treatment of de-
faulting debtors. Imprisonment for debt was the order of the day, from the time of the Statute of Merchants
in 1285, until DickensвЂ™ time in the mid-nineteenth century. The common law writs of capias authorized вЂњbody
execution,вЂќ i.e., seizure of the body of the debtor, to be held until payment of the debt.
English law was not unique in its lack of solicitude for debtors. HistoryвЂ™s annals are replete with tales of
draconian treatment of debtors. Punishments inп¬‚icted upon debtors included forfeiture of all property, re-
linquishment of the consortium of a spouse, imprisonment, and death. In Rome, creditors were apparently
authorized to carve up the body of the debtor, although scholars debate the extent to which the letter of that
law was actually enforced.
Direct Source: Charles Jordan Tabb, 1995, вЂњThe History of the Bankruptcy laws in the United States.вЂќ
www.bankruptcyп¬Ѓnder.com/historyofbkinusa.html. (The original article contains many more juicy historical
tidbits.)
п¬Ѓle=uncertainty.tex: RP
97
Section 5В·3. Uncertainty in Capital Budgeting, Debt, and Equity.

Table, and write down
an appropriate interest rate, given credit risk (from Section 5В·2). Start with the payoп¬Ђ table in
payoffs to project
Table 5.5. The creditor receives the property worth \$20,000 if the tornado strikes, or the full вЂњMortgage Lending.вЂќ
promised amount (to be determined) if the sun shines. To break even, the creditor must solve
for the payoп¬Ђ to be received if the sun will shine in exchange for lending \$25,000 today. This
is the вЂњquotedвЂќ or вЂњpromisedвЂќ payoп¬Ђ.
\$20, 000
= В·
\$25, 000 20%
1 + 10%
Promise
+ В·
80%
1 + 10%
(5.28)

+ Prob( Sunshine ) В· (Loan PV if Sunshine) .

Solving, the solution is a promise of
(1 + 10%) В· \$25, 000 в€’ 20% В· \$20, 000
Promise = = \$29, 375
80%
(5.29)
[1 + E (r )] В· Loan Value в€’ Prob(Tornado) В· Value if Tornado
=
Prob(Sunshine)

in repayment, paid by the borrower only if the sun will shine.
With this promised payoп¬Ђ of \$29,375 (if the sun will shine), the lenderвЂ™s rate of return will be The state-contingent
rates of return in the
the promised rate of return:
state and in the sunshine
\$29, 375 в€’ \$25, 000 state can be probability
rt=0,1 = = +17.50% , (5.30)
if Sunshine:
\$25, 000 weighted to arrive at the
expected rate of return.
The lender would not provide the mortgage at any lower promised interest rate. If the tornado
strikes, the owner walks away, and the lenderвЂ™s rate of return will be

\$20, 000 в€’ \$25, 000
rt=0,1 = = в€’20.00% . (5.31)
\$25, 000

Therefore, the lenderвЂ™s expected rate of return is

E (Лњt=0,1 ) = В·
r (в€’20.00%)
20%

+ В· = 10.00%
(+17.50%)
80%
(5.32)
r

+ Prob( Sunshine ) В· (rt=0,1 if Sunshine) .

After all, in a risk-neutral environment, anyone investing for one year expects to earn an ex-
pected rate of return of 10%.

The Levered Equity
Our interest now turns towards proper compensation for youвЂ”the expected payoп¬Ђs and ex- Now compute the
payoffs of the 60%
pected rate of return for you, the residual building owner. We already know the building
post-mortgage (i.e.,
is worth \$76,363.64 today. We also already know how the lender must be compensated: to levered) ownership of
contribute \$25,000 to the building price today, you must promise to pay the lender \$29,375 the building. The
method is exactly the
next year. Thus, as residual building owner, you need to pay \$51,363.64вЂ”presumably from
same.
personal savings. If the tornado strikes, these savings will be lost and the lender will repos-
sess the building. However, if the sun shines, the building will be worth \$100,000 minus the
promised \$29,375, or \$70,625. The ownerвЂ™s payoп¬Ђ table in Table 5.6 allows you to determine
п¬Ѓle=uncertainty.tex: LP
98 Chapter 5. Uncertainty, Default, and Risk.

that the expected future levered building ownership payoп¬Ђ is 20%В·\$0+80%В·\$70, 625 = \$56, 500.
Therefore, the present value of levered building ownership is

\$0 \$70, 625
PVt=0 = 20% В· + 80% В·
1 + 10% 1 + 10%
(5.33)
= Prob( Tornado ) В· (PV if Tornado) + Prob( Sunshine ) В· (PV if Sunshine)

= \$51, 363.64 .

Table 5.6. Payoп¬Ђ To Levered Building (Equity) Owner

Prob
Event Value Discount Factor
1/(1+10%)
1/(1+10%)
Sunshine 80% \$70,624.80

If the sun shines, the rate of return will be
Again, knowing the
state-contingent cash
\$70, 624.80 в€’ \$51, 363.63
п¬‚ows permits computing
rt=0,1 = = +37.50% . (5.34)
if Sunshine:
state-contingent rates of
\$51, 363.63
return and the expected
rate of return.
If the tornado strikes, the rate of return will be
\$0 в€’ \$51, 363.63
rt=0,1 = = в€’100.00% . (5.35)
\$51, 363.63

The expected rate of return of levered equity ownership, i.e., the building with the bundled
mortgage obligation, is

E (Лњt=0,1 ) = В·
r (в€’100.00%)
20%

+ В· = 10.00%
(+37.50%)
80%
(5.36)
r

+ Prob( Sunshine ) В· (rt=0,1 if Sunshine) .

Reп¬‚ections On The Example: Payoп¬Ђ Tables
Payoп¬Ђ tables are fundamental tools to think about projects and п¬Ѓnancial claims. You should
think about п¬Ѓnancial claims in terms of payoп¬Ђ tables.

Important: Whenever possible, in the presence of uncertainty, write down a
payoп¬Ђ table to describe the probabilities of each possible event (вЂњstateвЂќ) with its
state-contingent payoп¬Ђs.

Admittedly, this can sometimes be tedious, especially if there are many diп¬Ђerent possible states
(or even inп¬Ѓnitely many states, as in a bell-shaped normally distributed project outcomeвЂ”but
you can usually approximate even the most continuous and complex outcomes fairly well with
no more than ten discrete possible outcomes), but they always work!
п¬Ѓle=uncertainty.tex: RP
99
Section 5В·3. Uncertainty in Capital Budgeting, Debt, and Equity.

Table 5.7. Payoп¬Ђ Table and Overall Values and Returns

Prob
Event Building Value Mortgage Value Levered Ownership
Sunshine 80% \$100,000 \$29,375 \$70,625
Expected Valuet=1 \$84,000 \$27,500 \$56,500
PVt=0 \$76,364 \$25,000 \$51,364
E(rt=0,1 ) 10% 10% 10%

Table 5.7 shows how elegant such a table can be. It can describe everything we need in a There are three possible
investment
very concise manner: the state-contingent payoп¬Ђs, expected payoп¬Ђs, net present value, and
opportunities here. The
expected rates of return for your building scenario. Because owning the mortgage and the bank is just another
levered equity is the same as owning the full building, the last two columns must add up to the investor, with particular
payoff patterns.
values in the вЂњbuilding valueвЂќ column. You could decide to be any kind of investor: a creditor
(bank) who is loaning money in exchange for promised payment; a levered building owner who
is taking a вЂњpiece left over after a loanвЂќ; or an unlevered building owner who is investing money
into an unlevered project. You might take the whole piece (that is, 100% of the claimвЂ”all three
investments are just claims) or you might just invest, say \$5, at the appropriate fair rates of
return that are due to investors in our perfect world, where everything can be purchased at or
sold at a fair price. (Further remaining funds can be raised elsewhere.)

Reп¬‚ections On The Example and Debt and Equity Risk
We have not covered risk yet, because we did not need to. In a risk-neutral world, all that
matters is the expected rate of return, not how uncertain you are about what you will receive.
Of course, we can assess risk even in a risk-neutral world, even if risk were to earn no extra
So, which investment is most risky: full ownership, loan ownership, or levered ownership? Leveraging (mortgaging)
a project splits it into a
Figure 5.2 plots the histograms of the rates of return to each investment type. As the visual
safer loan and a riskier
shows, the loan is least risky, followed by the full ownership, followed by the levered ownership. levered ownership,
Your intuition should tell you that, by taking the mortgage, the medium-risky project вЂњbuildingвЂќ although everyone
has been split into a more risky project вЂњlevered buildingвЂќ and a less risky project вЂњmortgage.вЂќ
on average.
The combined вЂњfull building ownershipвЂќ project therefore has an average risk.
It should not come as a surprise to learn that all investment projects expect to earn a 10% If everyone is
risk-neutral, everyone
rate of return. After all, 10% is the time-premium for investing money. Recall from Page 92
should expect to earn
that the expected rate of return (the cost of capital) consists only of a time-premium and a 10%.
risk premium. (The default premium is a component only of promised interest rates, not of
expected interest rates; see Section 5В·2). By assuming that investors are risk-neutral, we have
assumed that the risk premium is zero. Investors are willing to take any investment that oп¬Ђers
an expected rate of return of 10%, regardless of risk.
Although our example has been a little sterile, because we assumed away risk preferences, it is Unrealistic, maybe! But
ultimately, maybe not.
nevertheless very useful. Almost all projects in the real world are п¬Ѓnanced with loans extended
by one party and levered ownership held by another party. Understanding debt and equity is
as important to corporations as it is to building owners. After all, stocks in corporations are
basically levered ownership claims that provide money only after the corporation has paid back
its loans. The building example has given you the skills to compute state-contingent, promised,
and expected payoп¬Ђs, and state-contingent, promised, and expected rates of returnsвЂ”the nec-
essary tools to work with debt, equity, or any other state-contingent claim. And really, all that
will happen later when we introduce risk aversion is that we will add a couple of extra basis
п¬Ѓle=uncertainty.tex: LP
100 Chapter 5. Uncertainty, Default, and Risk.

Figure 5.2. Probability Histograms of Project Returns

1.0
0.8
Histogram of Rate of Re-

0.6
Probability
turn for Project of Type

0.4
вЂњFull OwnershipвЂќ

0.2
0.0
в€’100 в€’50 0 50 100

Rate of Return, in %

1.0
0.8
Histogram of Rate of Re-

0.6
Probability
turn for Project of Type

0.4
вЂњLevered OwnershipвЂќ

0.2
(Most Risky)

0.0
в€’100 в€’50 0 50 100

Rate of Return, in %
1.0
0.8

Histogram of Rate of Re-
0.6
Probability

turn for Project of Type
0.4

вЂњLoan OwnershipвЂќ
0.2

(Least Risky)
0.0

в€’100 в€’50 0 50 100

Rate of Return, in %

20% probability 80% probability Expected

\$20, 000 \$29, 375 \$27, 500
Loan (Ownership) в€’1= в€’1= в€’1=
\$25, 000 \$25, 000 \$25, 000

в€’20.00% +17.50% в‰€ В±20%
10.00%
\$0 \$70, 625 \$56, 500
Levered Ownership в€’1= в€’1= в€’1=
\$51, 364 \$51, 364 \$51, 364

в€’100.00% +37.50% в‰€ В±60%
10.00%
\$20, 000 \$100, 000 \$84, 000
Full Ownership в€’1= в€’1= в€’1=
\$76, 364 \$76, 364 \$76, 364

в€’73.81% +30.95% в‰€ В±46%
10.00%

Consider the variability number here to be just an intuitive measureвЂ”it is the aforementioned вЂњstandard deviationвЂќ
that will be explained in great detail in Chapter 13. Here, it is computed as

20% В· (в€’20.00% в€’ 10%)2 + 80% В· (17.50% в€’ 10%)2
Sdv = = 19.94%
Loan Ownership

20% В· (в€’100.00% в€’ 10%)2 + 80% В· (37.50% в€’ 10%)2 = 59.04%
Levered Ownership Sdv =
(5.37)
20% В· (в€’73.81% в€’ 10%)2 + 80% В· (30.95% в€’ 10%)2
Sdv = = 46.07%
Full Ownership

Prob T В· [VT в€’ E (V )]2 + Prob S В· [VS в€’ E (V )]2
Sdv = .
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101
Section 5В·3. Uncertainty in Capital Budgeting, Debt, and Equity.

points of required compensationвЂ”more to equity (the riskiest claim) than to the project (the
medium-risk claim) than to debt (the safest claim).
Solve Now!
Q 5.14 In the example, the building was worth \$76,364, the mortgage was worth \$25,000, and
the equity was worth \$51,364. The mortgage thus п¬Ѓnanced 32.7% of the cost of the building, and
the equity п¬Ѓnanced \$67.3%. Is the arrangement identical to one in which two partners purchase
the building togetherвЂ”one puts in \$25,000 and owns 32.7%, and the other puts in \$51,364 and
owns 67.3%?

Q 5.15 Buildings are frequently п¬Ѓnanced with a mortgage that pays 80% of the price, not just
32.7% (\$25,000 of \$76,364). Produce a table similar to Table 5.7 in this case.

Q 5.16 Repeat the example if the loan does not provide \$25,000, but promises to pay oп¬Ђ \$25,000.
How much money do you get for this promise? What is the promised rate of return. How does the
riskiness of the project вЂњfull building ownershipвЂќ compare to the riskiness of the project вЂњlevered
building ownershipвЂќ?

Q 5.17 Repeat the example if the loan promises to pay oп¬Ђ \$20,000. Such a loan is risk-free. How
does the riskiness of the project вЂњfull building ownershipвЂќ compare to the riskiness of the project
вЂњlevered building ownershipвЂќ?

Q 5.18 Under risk-neutrality, a factory can be worth \$500,000 or \$1,000,000 in two years, de-
pending on product demand, each with equal probability. The appropriate cost of capital is 6%
per year. The factory can be п¬Ѓnanced with proceeds of \$500,000 from loans today. What are
the promised and expected cash п¬‚ows and rates of return for the factory (without loan), for the
loan, and for a hypothetical factory owner who has to п¬Ѓrst repay the loan?

Q 5.19 Advanced: For illustration, we assumed that the sample building was not lived in. It
consisted purely of capital amounts. But, in the real world, part of the return earned by a
building owner is rent. Now assume that rent of \$11,000 is paid strictly at year-end, and that
both the state of nature (tornado or sun) and the mortgage loan payment happens only after the
rent has been safely collected. The new building has a resale value of \$120,000 if the sun shines,
and a resale value of \$20,000 if the tornado strikes.

(a) What is the value of the building today?

(b) What is the promised interest rate for a lender providing \$25,000 in capital today?

(c) What is the value of residual ownership today?

(d) Conceptual Question: What is the value of the building if the owner chooses to live in the
building?
п¬Ѓle=uncertainty.tex: LP
102 Chapter 5. Uncertainty, Default, and Risk.

More Than Two Possible Outcomes
How does this example generalize to multiple possible outcomes? For example, assume that the
Multiple outcomes will
cause multiple building could be worth \$20,000, \$40,000, \$60,000, \$80,000, or \$100,000 with equal probability,
breakpoints.
and the appropriate expected interest rate were 10%вЂ”so the building has an PV of \$60, 000/(1+
10%) в‰€ \$54, 545.45. If a loan promised \$20,000, how much would you expect to receive? But,
of course, \$20,000!
E Payoп¬Ђ(Loan Promise =\$20,000) = \$20, 000 .
(5.38)
Payoп¬Ђ of Loan
E = .
Loan
if \$0 в‰¤ Loan в‰¤ \$20, 000
If a loan promised \$20,001, how much would you expect to receive? \$20,000 for sure, plus the
extra вЂњmarginalвЂќ \$1 with 80% probability. In fact, you would expect only 80 cents for each dollar
promised between \$20,000 and \$40,000. So, if a loan promised \$40,000, you would expect to

E = \$20, 000 + 80% В· (\$40, 000 в€’ \$20, 000)
Payoп¬Ђ( Loan Promise = \$40,000 )

(5.39)
= \$36, 000
Payoп¬Ђ of Loan
E = \$20, 000 + 80% В· (Loan в€’ \$20, 000) .
if \$20, 000 в‰¤ Loan в‰¤ \$40, 000

If a loan promised you \$40,001, how much would you expect to receive? You would get \$20,000
for sure, plus another \$20,000 with 80% probability (which is an expected \$16,000), plus the
marginal \$1 with only 60% probability. Thus,

E Payoп¬Ђ(Loan Promise = \$40,001) = + 80% В· (\$40, 000 в€’ \$20, 000)
\$20, 000

+ 60% В· \$1

(5.40)
= \$36, 000.60
Payoп¬Ђ of Loan
E = + 80% В· \$20, 000
\$20, 000
if \$40, 000 в‰¤ Loan в‰¤ \$60, 000
+ 60% В· (Loan в€’ \$40, 000) .

And so on. Figure 5.3 plots these expected payoп¬Ђs as a function of the promised payoп¬Ђs. With
this п¬Ѓgure, mortgage valuation becomes easy. For example, how much would the loan have to
promise to provide \$35,000 today? The expected payoп¬Ђ would have to be (1+10%)В·\$35, 000 =
\$38, 500. Figure 5.3 shows that an expected payoп¬Ђ of \$38,500 corresponds to around \$44,000
in promise. (The exact number can be worked out to be \$44,167.) Of course, we cannot borrow
more than \$54,545.45, the projectвЂ™s PV. So, we can forget about the idea of obtaining a \$55,000
mortgage.
Solve Now!
Q 5.20 What is the expected payoп¬Ђ if the promised payoп¬Ђ is \$45,000?

Q 5.21 What is the promised payoп¬Ђ if the expected payoп¬Ђ is \$45,000?

Q 5.22 Assume that the probabilities are not equal: \$20,000 with probability 12.5%, \$40,000
with probability 37.5%, \$60,000 with probability 37.5%, and \$80,000 with probability 12.5%.

(a) Draw a graph equivalent to Figure 5.3.

(b) If the promised payoп¬Ђ of a loan is \$50,000, what is the expected payoп¬Ђ?

(c) If the prevailing interest rate is 5% before loan payoп¬Ђ, then how much repayment does a
loan providing \$25,000 today have to promise? What is the interest rate?

You do not need to calculate these values, if you can read them oп¬Ђ your graph.
п¬Ѓle=uncertainty.tex: RP
103
Section 5В·3. Uncertainty in Capital Budgeting, Debt, and Equity.

Figure 5.3. Promised vs. Expected Payoп¬Ђs

60
50
40
Expect

30
20
10
0

0 20 40 60 80 100

Promise

Q 5.23 A new product may be a dud (20% probability), an average seller (70% probability) or
dynamite (10% probability). If it is a dud, the payoп¬Ђ will be \$20,000; if it is an average seller, the
payoп¬Ђ will be \$40,000; if it is dynamite, the payoп¬Ђ will be \$80,000. The appropriate expected
rate of return is 6% per year. If a loan promises to pay oп¬Ђ \$40,000, what are the promised and
expected rates of return?

Q 5.24 Advanced: What is the formula equivalent to (5.40) for promised payoп¬Ђs between \$60,000
and \$80,000?

Q 5.25 Advanced: Can you work out the exact \$44,167 promise for the \$35,000 (today!) loan?
п¬Ѓle=uncertainty.tex: LP
104 Chapter 5. Uncertainty, Default, and Risk.

Although it would be better to get everything perfect, it is often impossible to come up with
perfect cash п¬‚ow forecasts and appropriate interest rate estimates. Everyone makes errors.
So, how bad are mistakes? How robust is the NPV formula? Is it worse to commit an error in
estimating cash п¬‚ows or in estimating the cost of capital? To answer these questions, we will
do a simple form of scenario analysisвЂ”we will consider just a very simple project, and see
how changes in estimates matter to the ultimate value. Doing good scenario analysis is also
good practice for any managersвЂ”so that they can see how sensitive their estimated value is to
reasonable alternative possible outcomes. Therefore this method is also called a sensitivity
analysis. Doing such analysis becomes even more important when we consider вЂњreal optionsвЂќ
in our next chapter.

5В·4.A. Short-Term Projects

Assume that your project will pay oп¬Ђ \$200 next year, and the proper interest rate for such
The benchmark case: A
short-term project, projects is 8%. Thus, the correct project present value is
correctly valued.
\$200
(5.44)
PVcorrect = в‰€ \$185.19 .
1 + 8%

If you make a 10% error in your cash п¬‚ow, e.g., mistakenly believing it to return \$220, you will
Committing an error in
cash п¬‚ow estimation. compute the present value to be

\$220
(5.45)
PVCF error = в‰€ \$203.70 .
1 + 8%
The diп¬Ђerence between \$203.70 and \$185.19 is a 10% error in your present value.
In contrast, if you make a 10% error in your cost of capital (interest rate), mistakenly believing
Committing an error in
interest rate estimation. it to require a cost of capital (expected interest rate) of 8.8% rather than 8%, you will compute
the present value to be
\$200
(5.46)
= в‰€ \$183.82 .
PVr error
1 + 8.8%
The diп¬Ђerence between \$183.82 and \$185.19 is less than a \$2 or 1% error.

Important: For short-term projects, errors in estimating correct interest rates
are less problematic in computing NPV than are errors in estimating future cash
п¬‚ows.

5В·4.B. Long-Term Projects

Now take the same example, but assume the cash п¬‚ow will occur in 30 years. The correct
A long-term project,
correctly valued and present value is now
incorrectly valued.
\$200
PVcorrect = в‰€ \$19.88 (5.47)
.
(1 + 8%)30
The 10% вЂњcash п¬‚ow errorвЂќ present value is

\$220
PVCF error = в‰€ \$21.86 (5.48)
,
(1 + 8%)30

and the 10% вЂњinterest rate errorвЂќ present value is

\$200
= в‰€ \$15.93 (5.49)
.
PVr error
(1 + 8.8%)30
п¬Ѓle=uncertainty.tex: RP
105

This calculation shows that cash п¬‚ow estimation errors and interest rate estimation errors are Both cash п¬‚ow and cost
of capital errors are now
now both important. So, for longer-term projects, estimating the correct interest rate becomes
important.
relatively more important. However, in fairness, estimating cash п¬‚ows thirty years into the
future is just about as diп¬ѓcult as reading a crystal ball. (In contrast, the uncertainty about the
long-term cost of capital tends not explode as quickly as your uncertainty about cash п¬‚ows.)
Of course, as diп¬ѓcult as it may be, we have no alternative. We must simply try to do our best
at forecasting.

Important: For long-term projects, errors in estimating correct interest rates
and errors in estimating future cash п¬‚ows are both problematic in computing NPV.

5В·4.C. Two Wrongs Do Not Make One Right

Please do not think that you can arbitrarily adjust an expected cash п¬‚ow to paint over an issue Do not think two errors
cancel.
with your discount rate estimation, or vice-versa. For example, let us presume that you consider
an investment project that is a bond issued by someone else. You know the bondвЂ™s promised
payoп¬Ђs. You know these are higher than the expected cash п¬‚ows. Maybe you can simply use
the average promised discount rate on other risky bonds to discount the bondвЂ™s promised cash
п¬‚ows? After all, the latter also reп¬‚ects default risk. The two default issues might cancel one
another, and you might end up with the correct number. Or they might not cancel and you end
up with a non-sense number!
LetвЂ™s say the appropriate expected rate of return is 10%. A suggested bond investment may An example: do not
think you can just work
promise \$16,000 for a \$100,000 investment, but have a default risk on the interest of 50% (the
with promised rates.
principal is insured). Your benchmark promised opportunity cost of capital may rely on risky
bonds that have default premia of 2%. Your project NPV is neither в€’\$100, 000 + \$116, 000/(1 +
12%) в‰€ \$3, 571 nor в€’\$100, 000 + \$100, 000/(1 + 10%) + \$16, 000/(1 + 12%) в‰€ \$5, 195. Instead,
you must work with expected values

\$100, 000 \$8, 000
(5.50)
PV = в€’\$100, 000 + + в‰€ в€’\$1, 828 .
1 + 10% 1 + 10%
This bond would be a bad investment.
Solve Now!
Q 5.26 What is the relative importance of cash п¬‚ow and interest rate errors for a 10-year project?

Q 5.27 What is the relative importance of cash п¬‚ow and interest rate errors for a 100-year
project?
п¬Ѓle=uncertainty.tex: LP
106 Chapter 5. Uncertainty, Default, and Risk.

5В·5. Summary

The chapter covered the following major points:

вЂў The possibility of future default causes promised interest rates to be higher than expected
interest rates. Default risk is also often called credit risk.

вЂў Quoted interest rates are almost always promised interest rates, and are higher than
expected interest rates.

вЂў Most of the diп¬Ђerence between promised and expected interest rates is due to default.
Extra compensation for bearing more riskвЂ”the risk premiumвЂ”is typically much smaller

вЂў The key tool for thinking about uncertainty is the payoп¬Ђ table. Each row represents one
possible state outcome, which contains the probability that the state will come about, the
total project value that can be distributed, and the allocation of this total project value
to diп¬Ђerent state-contingent claims. The state-contingent claims вЂњcarve upвЂќ the possible
project payoп¬Ђs.

вЂў Most real-world projects are п¬Ѓnanced with the two most common state-contingent claimsвЂ”
debt and equity. The conceptual basis of debt and equity is п¬Ѓrmly grounded in payoп¬Ђ
tables. Debt п¬Ѓnancing is the safer investment. Equity п¬Ѓnancing is the riskier investment.

вЂў If debt promises to pay more than the project can deliver in the worst state of nature, then
the debt is risky and requires a promised interest rate in excess of its expected interest
rate.

вЂў NPV is robust to uncertainty about the expected interest rate (the discount rate) for short-
term projects. However, NPV is not robust with respect to either expected cash п¬‚ows or
discount rates for long-run projects.
п¬Ѓle=uncertainty.tex: RP
107
Section 5В·5. Summary.

Solutions and Exercises

1. No! It is presumed to be knownвЂ”at least for a die throw. The following is almost philosophy and beyond
what you are supposed to know or answer here: It might, however, be that the expected value of an investment
is not really known. In this case, it, too, could be a random variable in one senseвЂ”although you are presumed
to be able to form an expectation (opinion) over anything, so in this sense, it would not be a random variable,
either.
2. Yes and no. If you do not know the exact bet, you may not know the expected value.
3. If the random variable is the number of dots on the die, then the expected outcome is 3.5. The realization
was 6.
4. The expected value of the stock is \$52. Therefore, purchasing the stock at \$50 is not a fair bet, but a good
bet.

5. Only for government bonds. Most other bonds have some kind of default risk.
6.
(a) The expected payoп¬Ђ is now 95%В·\$210 + 1%В·\$100 + 4%В·\$0 = \$200.50. Therefore, the expected rate of
return is \$200.50/\$200 = 0.25%.
(b) You require an expected payoп¬Ђ of \$210. Therefore, you must solve for a promised payment 95%В·P +
1%В·\$100 + 4%В·\$0 = \$210 в†’ P = \$209/0.95 = \$220. On a loan of \$200, this is a 10% promised interest
rate.
7. The expected payoп¬Ђ is \$203.50, the promised payoп¬Ђ is \$210, and the stated price is \$210/(1+12%)=\$187.50.
The expected rate of return is \$203.50/\$187.50 = 8.5%. Given that the time premium, the Treasury rate is 5%,
the risk premium is 3.5%. The remaining 12%-8.5%=3.5% is the default premium.

8. The actual cash п¬‚ow is replaced by the expected cash п¬‚ow, and the actual rate of return is replaced by the
expected rate of return.
9. \$750, 000/(1 + 6%)2 в‰€ \$667, 497.33.
10. E (P ) = 20% В· \$20, 000 + 70% В· \$40, 000 + 10% В· \$80, 000 = \$40, 000.
11. \$37, 037.
12. \$20, 000/\$37, 037 в€’ 1 = в€’46%, \$40, 000/\$37, 037 в€’ 1 = +8%, \$80, 0000/\$37037 в€’ 1 = +116%.
13. 20%В·(в€’46%) + 70%В·(+8%) + 10%В·(+116%) = 8%.
14. No! Partners would share payoп¬Ђs proportionally, not according to вЂњdebt comes п¬Ѓrst.вЂќ For example, in the
tornado state, the 32.7% partner would receive only \$6,547.50, not the entire \$20,000 that the debt owner
15. The mortgage would п¬Ѓnance \$61,090.91 today.
Prob
Event Building Value Mortgage Value Levered Ownership
Sunshine 80% \$100,000 \$79,000 \$21,000
Expected Valuet=1 \$84,000 \$67,200 \$16,800
PVt=0 \$76,364 \$61,091 \$15,273
E (rt=0,1 ) 10% 10% 10%
16. In the tornado state, the creditor gets all (\$20,000). In the sunshine state, the creditor receives the promise
of \$25, 000. Therefore, the creditorвЂ™s expected payoп¬Ђ is 20% В· \$20, 000 + 80% В· \$25, 000 = \$24, 000. To oп¬Ђer
an expected rate of return of 10%, you can get \$24, 000/1.1 = \$21, 818 from the creditor today. The promised
rate of return is therefore \$25, 000/\$21, 818 в€’ 1 = 14.58%.
17. The loan pays oп¬Ђ \$20,000 for certain. The levered ownership pays either \$0 or \$80,000, and costs \$64, 000/(1+
10%) = \$58, 182. Therefore, the rate of return is either в€’100% or +37.5%. We have already worked out full
ownership. It pays either \$20,000 or \$100,000, costs \$76,364, and oп¬Ђers either в€’73.81% or +30.95%. By
inspection, the levered equity project is riskier. In eп¬Ђect, building ownership has become riskier, because the
owner has chosen to sell oп¬Ђ the risk-free component, and retain only the risky component.
18. Factory: The expected factory value is \$750,000. Its price would be \$750, 000/1.062 = \$667, 497. The
promised rate of return is therefore \$1, 000, 000/\$667, 497 в€’ 1 в‰€ 49.8%. Loan: The discounted (todayвЂ™s) loan
price is \$750, 000/1.062 = \$667, 497.33. The promised value is \$1,000,000. The loan must have an expected
payoп¬Ђ of 1.062 В· \$500, 000 = \$561, 800 (6% expected rate of return, two years). Because the loan can pay
\$500,000 with probability 1/2, it must pay \$623,600 with probability 1/2 to reach \$561,800 as an average.
Therefore, the promised loan rate of return is \$623, 600/\$500, 000 в€’ 1 = 24.72% over two years (11.68%
per annum). Equity: The levered equity must therefore pay for/be worth \$667, 497.33 в€’ \$500, 000.00 =
\$167, 497.33 (alternatively, levered equity will receive \$1, 000, 000 в€’ \$623, 600 = \$376, 400 with probability
1/2 and \$0 with probability 1/2), for an expected payoп¬Ђ of \$188,200. (The expected two-year holding rate
of return is \$188, 200/\$167, 497 в€’ 1 = 12.36% [6% per annum, expected].) The promised rate of return is
(\$1, 000, 000 в€’ \$623, 600)/\$167, 497.33 в€’ 1 = 124.72% (50% promised per annum).
п¬Ѓle=uncertainty.tex: LP
108 Chapter 5. Uncertainty, Default, and Risk.

19.
(a) In the sun state, the value is \$120,000+\$11,000= \$131,000. In the tornado state, the value is \$11,000+\$20,000=
\$31,000. Therefore, the expected building value is \$111,000. The discounted building value today is
\$100,909.09.
(b) Still the same as in the text: the lenderвЂ™s \$25,000 loan can still only get \$20,000, so it is a promise for
\$29,375. So the quoted interest rate is still 17.50%.
(c) \$100, 909.09 в€’ \$25, 000 = \$75, 909.09.
(d) Still \$100,909.09, assuming that the owner values living in the building as much as a tenant would.
Owner-consumed rent is the equivalent of corporate dividends paid out to levered equity. Note: you can repeat
this example assuming that the rent is an annuity of \$1,000 each month, and tornadoes strike mid-year.
20. From the graph, it is around \$40,000. The correct value can be obtained by plugging into Formula (5.40):
\$39,000.
21. From the graph, it is around \$55,000. The correct value can be obtained by setting Formula (5.40) equal to
\$55,000 and solving for вЂњLoan.вЂќ The answer is indeed \$55,000.
22.
60
50
40
Expect

30
20
10
0

0 20 40 60 80 100

Promise
(a)
(b) The exact expected payoп¬Ђ is 1/8 В· \$20, 000 + 3/8 В· \$40, 000 + 1/2 В· \$50, 000 = \$42, 500. The 1/2 is the
probability that you will receive the \$50,000 that you have been promised, which occurs if the project
ends up worth at least as much as your promised \$50,000. This means that it is the total probability
that it will be worth \$60,000 or \$80,000.
(c) The loan must expect to pay oп¬Ђ (1 + 5%) В· \$25, 000 = \$26, 250. Therefore, solve 1/8 В· \$20, 000 + 7/8 В· x =
\$26, 250, so the exact promised payoп¬Ђ must be x = \$27, 142.90.

23. With 20% probability, the loan will pay oп¬Ђ \$20,000; with 80% probability, the loan will pay oп¬Ђ the full promised
\$40,000. Therefore, the loanвЂ™s expected payoп¬Ђ is 20%В·\$20, 000 + 80%В·\$40, 000 = \$36, 000. The loanвЂ™s price
is \$36, 000/(1 + 6%) = \$33, 962. Therefore, the promised rate of return is \$40, 000/\$33, 962 в€’ 1 в‰€ 17.8%. The
expected rate of return was given: 6%.
24.
Payoп¬Ђ of Loan
E = \$20, 000
if \$60, 000 в‰¤ Loan в‰¤ \$80, 000
+ 80% В· \$20, 000
(5.51)
+ 60% В· \$20, 000

+ 40% В· (Loan в€’ \$60, 000) .

25. The loan must yield an expected value of \$38,500. Set formula (5.40) equal to \$38,500 and solve for вЂњLoan.вЂќ
п¬Ѓle=uncertainty.tex: RP
109
Section 5В·5. Summary.

26. Consider a project that earns \$100 in 10 years, and where the correct interest rate is 10%.
вЂў The correct PV is \$100/(1 + 10%)10 = \$38.55.
вЂў If the cash п¬‚ow is incorrectly estimated to be 10% higher, the incorrect PV is \$110/(1 + 10%)10 = \$42.41.
вЂў If the interest rate is incorrectly estimate to be 10% lower, the incorrect PV is \$100/(1 + 9%)10 = \$42.24.
So, the misvaluation eп¬Ђects are reasonably similar at 10% interest rates. Naturally, percent valuation mistakes
in interest rates are higher if the interest rate is higher; and lower if the interest rate is lower.
27. Although this, too, depends on the interest rate, interest rate errors almost surely matter for any reasonable
interest rates now.

(All answers should be treated as suspect. They have only been sketched, and not been checked.)
п¬Ѓle=uncertainty.tex: LP
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